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A polygon has exactly 93 sides. If the measure of each of the 93 interior angles of this polygon is [tex](180p)^\circ[/tex], what is the value of [tex]p[/tex]?

Answer :

To find the value of [tex]p[/tex] in the given problem, we need to understand the relationship between the number of sides in a polygon and the measure of its interior angles.


  1. Formula for Interior Angles:
    The sum of the interior angles [tex]S[/tex] of a polygon with [tex]n[/tex] sides is given by the formula:
    [tex]S = (n - 2) \times 180°[/tex]
    where [tex]n[/tex] is the number of sides of the polygon.


  2. Sum of Interior Angles for a 93-sided Polygon:
    For a polygon with 93 sides ([tex]n = 93[/tex]):
    [tex]S = (93 - 2) \times 180°[/tex]
    [tex]S = 91 \times 180°[/tex]


  3. Measure of Each Interior Angle:
    Since the problem states that each interior angle measures [tex]180p[/tex], we first find the measure of a single interior angle by dividing the sum by the number of sides:
    [tex]\text{Each Angle} = \frac{S}{93} = \frac{91 \times 180°}{93}[/tex]


  4. Calculating the Expression:
    Simplify the fraction:
    [tex]\text{Each Angle} = \frac{16380}{93}[/tex]
    Simplifying this expression:
    [tex]\text{Each Angle} = 176.129... \text{approximately}[/tex]


  5. Setting the Equation:
    Since each angle is [tex]180p[/tex], we set up the equation:
    [tex]180p = 176.129...[/tex]


  6. Finding [tex]p[/tex]:
    Solving for [tex]p[/tex]:
    [tex]p = \frac{176.129}{180}[/tex]
    [tex]p \approx 0.9785[/tex]



Therefore, the value of [tex]p[/tex] is approximately 0.9785. This shows how changes in the number of polygon sides affect each angle's measure.

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